Here we study a fairly general jump–diffusion price process. We investigate the existence of equivalent martingale meaures, derive the Hansen–Jagannathan bounds, and extend the theory to include dividends. Completeness questions are discussed in some detail, and we also develop the theory for change of numeraire.

This chapter is devoted to the classical Merton jump–diffusion model which is analyzed using martingale methods. Given the standard Merton assumption, we derive pricing formulas for a standard call option as well as for an asset-or-nothing option.

In this chapter we present an approach to pricing in incomplete markets where, instead of looking for a unique price, we look for "reasonable" pricing bounds. The term reasonable is formalized as a bound on the market price of risk, and it turns out that the "good-deal bounds" can be computed by solving a standard stochastic control problem. The theory is then applied to an example.

In this chapter we extend the earlier filtering results to k-variate and to MPP observations.

In this chapter we present a self-contained chapter on dynamic programming in continuous time in the framework of jump diffusions driven by a marked point process. We derive the relevant HJB equation and we study some examples of standard control as well as intensity control.

In this chapter we study a large market with diversifiable jump risk. The question to solve is to see whether no arbitrage implies that the diversifiable risk is not priced by the market. It turns out that the answer is yes, but only asymptotically.

In this chapter we study dynamic market equilibrium. This is done within the framework of a representative agent and a unit net supply jump–diffusion endowment process. We derive formulas for the equilibrium short, the equilibrium stochastic discount factor, and the equilibrium market prices of risk. We also study factor models and give concrete examples. We finish by studying an equilibrium model with partial observations, using our earlier filtering results.

In this chapter we discuss the various possibilities of pricing that are at hand when the market is incomplete. This includes topics such as the Esscher transformation, f-divergences, the minimal martingale measure, and utility indifference pricing.

This is the start of a part of the book devoted to non-linear filtering with Wiener and point-process observations. This chapter deals with filtering with Wiener noise and we derive the Fujisaki–Kallinapur–Kunita filtering equations. We discuss finite-dimensional filters and we derive the Kalman and Wonham filters.

This chapter is devoted to the connection between stochastic differential equations and partial integro-differential equations. We discuss and derive the infinitesimal generator for a jump diffusion. We then derive the Kolmogorov backward equation as well as the Feynman–Kac representation.

This chapter studies interest theory within a jump–diffusion framework. We start by studying short-rate models and affine term structures. We then go on to forward-rate models of HJM type, and for these models we derive the relevant HJM drift condition, guaranteeing the existence of a martingale measure. We also briefly discuss infinite-dimensional SDEs and the Musiela parameterization of the forward-rate curve.

The concept of a stochastic intensity for a counting process is introduced. The interpretation is discussed in some detail. Existence and uniqueness is treated and we investigate the dependence upon the filtration.

This chapter contains an introduction to financial economics, giving the reader the necessary background for the rest of the text. It coversportfolio theory, arbitrage theory, martingale measures, change of numeraire, stochastic discount factors, Hansen–Jagannathan bounds, dividends and consumption.

We introduce counting processes, and we discuss the Poisson process and its intensity in some detail.